Properties

Label 199920cy
Number of curves $6$
Conductor $199920$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cy1")
 
E.isogeny_class()
 

Elliptic curves in class 199920cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
199920.c5 199920cy1 \([0, -1, 0, 2399024, 1388370880]\) \(3168685387909439/3563732336640\) \(-1717328059078079938560\) \([2]\) \(10616832\) \(2.7610\) \(\Gamma_0(N)\)-optimal
199920.c4 199920cy2 \([0, -1, 0, -13657296, 13090216896]\) \(584614687782041281/184812061593600\) \(89059140544206628454400\) \([2, 2]\) \(21233664\) \(3.1076\)  
199920.c2 199920cy3 \([0, -1, 0, -198054096, 1072707988416]\) \(1782900110862842086081/328139630024640\) \(158127306067021297090560\) \([2]\) \(42467328\) \(3.4542\)  
199920.c3 199920cy4 \([0, -1, 0, -86161616, -297866310720]\) \(146796951366228945601/5397929064360000\) \(2601209677794875965440000\) \([2, 2]\) \(42467328\) \(3.4542\)  
199920.c6 199920cy5 \([0, -1, 0, 33790384, -1062392377920]\) \(8854313460877886399/1016927675429790600\) \(-490047586658875122890342400\) \([2]\) \(84934656\) \(3.8008\)  
199920.c1 199920cy6 \([0, -1, 0, -1366182736, -19435718080064]\) \(585196747116290735872321/836876053125000\) \(403282455650726400000000\) \([2]\) \(84934656\) \(3.8008\)  

Rank

sage: E.rank()
 

The elliptic curves in class 199920cy have rank \(1\).

Complex multiplication

The elliptic curves in class 199920cy do not have complex multiplication.

Modular form 199920.2.a.cy

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4 q^{11} - 6 q^{13} + q^{15} - q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.